Bracket Walk

AtCoder

IDarc173_d

Time3000ms

Memory256MB

Difficulty—

You are given a directed graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered from $1$ to $N$, and each edge is labeled with `(` or `)`. The $i$\-th edge is directed from vertex $u_i$ to vertex $v_i$ with a label $c_i$. The graph does not contain multi-edges or self-loops. In this graph, for any two vertices $s$ and $t$, there is a path from $s$ to $t$. Determine if there is a **walk** on the graph $G$ that satisfies all of the following conditions: * The start and end vertices of the walk are the same. * For $i=1,2,\dots,M$, the $i$\-th edge is used at least once in the walk. * The string obtained by arranging the labels of the edges used in the walk in the order of their usage is a regular bracket sequence. What is a walk? A walk on a graph $G$ is a sequence $(v_1,e_1,v_2,\dots,v_{k-1},e_{k-1},v_k)$ of $k$ vertices ($k$ is a positive integer) and $k-1$ edges, where the edge $e_i$ is directed from vertex $v_i$ to vertex $v_{i+1}$. The vertices $v_1$ and $v_k$ are called the start and end vertices of the walk, respectively. What is a regular bracket sequence?A regular bracket sequence is a string that satisfies one of the following conditions: * It is an empty string. * It is a string obtained by concatenating `(`, a regular bracket sequence $A$, and `)` in this order. * It is a string obtained by concatenating two non-empty regular bracket sequences $A$ and $B$ in this order. ## Constraints * $2 \leq N \leq 4000$ * $N \leq M \leq 8000$ * $1 \leq u_i,v_i \leq N$ * $c_i$ is `(` or `)`. * $u_i \neq v_i$ * If $i \neq j$, then $(u_i,v_i) \neq (u_j,v_j)$. * All input values are integers. * In the input graph, for any two vertices $s$ and $t$, there is a path from $s$ to $t$. ## Input The input is given from Standard Input in the following format: $N$ $M$ $u_1$ $v_1$ $c_1$ $u_2$ $v_2$ $c_2$ $\vdots$ $u_M$ $v_M$ $c_M$ [samples]

Samples

Input #1

Output #1

Yes

The walk that uses edges $1,2,3,4,1,5,6,7$ in this order uses all the edges at least once, and the string `()()()()` obtained by arranging the labels of the edges in the order of their usage is a regular bracket sequence, so all conditions are satisfied.
The walk may use the same edge multiple times or visit the same vertex multiple times.

Input #2

2 2
1 2 )
2 1 )

Output #2

No

Input #3

Output #3

Yes

API Response (JSON)

{
  "problem": {
    "name": "Bracket Walk",
    "description": {
      "content": "You are given a directed graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered from $1$ to $N$, and each edge is labeled with `(` or `)`. The $i$\\-th edge is directed from vertex $u_i$ ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc173_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a directed graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered from $1$ to $N$, and each edge is labeled with `(` or `)`. The $i$\\-th edge is directed from vertex $u_i$ ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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